I'm just getting acquainted with OSM (OpenStreetMap), so this might be a total newbie question, but I'm looking for OSM Point of Interest data for all metro stations in Paris. I'm sure this info should be available, as maps I've seen maps of Paris using OSM data (e.g., mapbox.com default maps) display these metro stations, but I'm not sure where to look.

Any pointers for downloading this data would be highly appreciated.

After reviewing the answers provided and doing some digging myself, my conclusions are:

Using Overpass API seems to be the most user-friendly as far as I can tell (when you get the hang of the query language, that is). For example, the query below returns the actual routing (metro lines) plus the referenced metro stations in nice JSON format.

One other approach is to use the Overpass javascript library to query the read-only OSM API. You can write a query that will select all of the relevant tags for an area. Here is an example from the wiki:

The key part is the station + railway part. If you export a large enough area (or use a pre-made export, as suggested by Fezter) and query for that tag, you should get every node that is a railway station. If you only want metro stations, then you might want to filter by other tags as well.